_{1}

^{*}

This paper contrasts high-risk, hedge fund trading, with low-risk, mutual fund trading, in terms of their differing utility functions. We envision hedge funds, led by informed traders who use information to seek out investment opportunities, timing market conditions, with the expectation that prices will move in their favor. Directional hedge funds act to influence prices, while non-directional hedge funds do not act to influence prices. We present utility functions based on steeply-sloping Laplace distributions and hyperbolic cosine distributions, to describe the actions of directional hedge fund traders. Less steeply-sloping lognormal distributions, Coulomb wave functions, quadratic utility functions, and Bessel utility functions are used to describe the investing style of non-directional hedge fund traders. Flatter Legendre utility functions and inverse sine utility functions describe the modest profit-making aspirations of mutual fund traders. The paper’s chief contribution is to develop optimal prices quantitatively, by intersecting utility functions with price distributions. Price distributions for directional hedge fund returns are portrayed as sharp increases and decreases, in the form of jumps, in a discrete arrival Poisson-distributed process. Separate equations are developed for direc tional hedge fund strategies, including event-driven arbitrage, and global ma cro strategies. Non-directional strategies include commodity trading, risk-neutral arbitrage, and convertible arbitrage, with primarily lognormal pricing distributions, and some Poisson jumps. Mutual funds are perceived to be Markowitz portfolios, lying on the Capital Market Line, or the International Capital Market Line, tangent to the Efficient Frontier of minimum variance-maximum return portfolios.

Hedge funds, which attract large cash inflows per investor, operate with minimum regulations, and strict entry barriers. Leverage, arbitrage, short selling, and derivative strategies are permitted. Only high net worth investors are included. We consider hedge funds to engage in some combination of three activities, i.e. actively seeking out investment opportunities, predicting market conditions through timing strategies, or herding to influence prices. Directional hedge funds act to move prices, while non-directional hedge funds position themselves to take advantage of information about market conditions, without actively trading to influence prices. For example, directional hedge funds predict that certain mergers are not going to be completed. They purchase the target stock, using timing to predict the date of announcement of deal failure. They short sell the stock just prior to the announcement of deal failure, earning significant gains. A non-directional hedge fund strategy could be the carry trade of borrowing funds at low-interest rates in US dollars, say at 3%, to invest in high-interest Australian dollars, at about 8%. Hedge funds have experienced record cash inflows in recent years, such as $3.21 trillion in 2017 [

The core distinction between hedge fund investing and mutual fund investing lies in the attitude toward risk. While hedge fund investing uses risk-taking to generate abnormal profits, mutual fund investing is restricted by corporate practice, to modest, low-risk returns. The academic literature contrasts these risk-taking and risk averse strategies. [

The purpose of this paper is to 1) Relax the assumption of isoelastic utility that all investors have identical utility functions. 2) Specify hedge fund pricing distributions, based on Poisson jump processes, and 3) Evaluate the level of risk aversion in each investing strategy. Unlimited profit potential is sought in hedge fund investing, with highly uncertain, time-specific investments. Such upside potential does not exist for mutual funds.

The remainder of this paper is organized as follows. Section 2 consists of a Review of Literature. Section 3 develops the Proposed Quantitative Formulations of Hedge Fund Pricing, and Mutual Fund Pricing, while Section 4 provides Conclusions, and Recommendations for Future Research.

Hedge fund traders engage in directional, or non-directional, informed trading. Applying [

Mutual fund investing originates from the creation of minimum-variance portfolios. Their objective is to minimize risk for a certain level of return. This is a risk-averse strategy, as additional risk is only accepted if additional return is guaranteed. [

R p = R f + ( R M − R f ) σ P / σ M (1)

R = return on the mutual fund, ( R M − R f ) = market risk premium, σ P , σ M = standard deviation of the mutual fund and market portfolios respectively [

Mutual fund managers are liquidity traders, who engage in purchasing and selling stock, with the view to increasing their inventory of stock. They do not take advantage of price run-ups prior to information events, such as earnings surprises. There is no timing to predict the best time to buy or sell, or herding, to increase or decrease prices. They trade at prevailing bid and ask prices, set by market makers. Market makers do not adjust prices, as the lack of herding means that only a few traders demand stock at any time, so that prevailing prices do not need to be adjusted for excess demand [

The Miller-Swanson Schema may be used to differentiate between the attitudes of hedge fund traders and mutual fund traders. The Schema’s entrepreneurial orientation describes individuals driven by the desire to manipulate risk to maximize income, which we may liken to the mores of hedge fund managers. This personality is contrasted with the mutual fund manager’s bureaucratic orientation of favoring a low-risk, stable relationship with an employer [

Y 1 = Y 0 + r e Y 0 (2)

The initial growth rate, Y_{0}, grows by a variable growth rate, r_{e}, to yield the final growth rate, Y_{1}.

For the bureaucratic orientation, the end-period income is,

Y 1 = Y 0 + r b Y 0 (3)

The initial growth rate, Y_{0}, grows by a fixed growth rate, r_{b}, to yield the final growth rate, Y_{1}.

[

Initial formulations of hedge fund models employed the mean-variance framework [

max w T μ st w T Σ w ≤ σ max 2 (4)

or,

max w T μ − ( ƛ / 2 ) w T Σ w (5)

where, ƛ = Lagrange multiplier, coefficient of absolute risk aversion In Equation (4), the upper limit on risk is σ_{max}, which limits the level of risk in the mean-variance portfolio. Equation (5) suggests that the maximization of portfolio return may be restricted by risk, so that investors have an incentive to reduce risk in order to achieve maximum returns. This condition is violated by hedge funds. Hedge fund portfolios have utility functions that support the achievement of a threshold return, followed by the pursuit of unlimited profits. [

E [ U ( W T w 1 ) ] > E [ U ( W T w 2 ) ] (6)

A portfolio, w_{1}, with higher expectations of random wealth, W T w 1 is preferred to portfolio w_{2}, with lower expectations of random wealth, W T w 2 . [

returns of a hedge fund over the risk-free rate, as the product of global risk aversion, − E u ″ ( 1 + r p ) / E u ′ ( 1 + r p ) and a total portfolio risk measure, termed omega, C o v ( r p , u ′ ( 1 + r p ) ) / [ E u ″ ( 1 + r p ) ] , where r_{p} is the risk of the portfolio.

Hedge fund returns increased with the decrease in overall risk aversion, and increased with the extent to which hedge fund risk varied with the risk of the remainder of the portfolio, the path of variation of returns following a gamma distribution. Alternatively, [

U ( x ) = x + H + ( 1 − Ώ ( H ) ) ( H − x ) (7)

H = threshold, and Ώ = omega ratio. [_{f}, returns, (u − r_{f}) on X_{0} shares of a risky asset, and returns on a call option, X 1 ∗ ( R c − c ⋅ r f ) , where X_{1} = number of call options, and ( R c − c ⋅ r f ) = gain per call. While these formulations provide descriptions of utility functions, they fail to provide a theoretical framework to account for hedge fund trading.

A useful distinction between hedge fund traders and mutual fund traders lies in our application of [

Intertemporal utility functions view utility as being derived from two or more periods. [_{1}), that describes the dissatisfaction with Period 1 consumption is,

U ( x 1 ) = x 1 1 − η / ( 1 − η ) + e − δ / ( 1 − η ) [ E p x / 2 − R R A ] ^ ( 1 − η ) / ( 1 − R R A ) (8)

where δ = risk preference, E p = expected return on portfolio, p, RRA = coefficient of relative risk aversion, with higher risk aversion in Period 1, over Period 2.

The limitation of Equation (8) is that it does not distinguish between risk-taking hedge funds and risk averse mutual funds, the only separability being between period 1, and period 2. [

[

In summary, this paper views the above studies as providing only partial formulations of mutual fund and hedge fund distributions, as the distributions mentioned are pricing distributions. There is no mention of utility functions intersecting with pricing distributions to yield optimal prices, an omission that this paper rectifies in Section 3. Also mutual fund studies assume that traders base returns on market outcomes, only. Stocks of multinationals are highly sensitive to exchange rates, tariffs, and events, such as Brexit. Returns on mutual funds vary widely from market returns during periods of global macroeconomic instability. Sector funds follow a particular industry sector, not the broad market. In Section 3, this paper provides a variety of formulations of both mutual fund strategies and hedge fund strategies, with a view toward developing a comprehensive framework.

Event-Driven Arbitrage. We will use price run-ups on stock mergers as a case of directional hedge fund trading. [

See

n [ 2 e 3 a ( 1 − a ) − 2 e a ( 1 + a ) / 2 a e a ( e 2 a + ∑ ( 1 − e − ƛ x i sinh a ( 1 − e − ƛ x i ) / cosh a ( 1 − e − ƛ x i ) = 1 / √ Π η 2 e ^ − ( y − δ ) 2 / 2 η 2 d y (9)

The derivative of Equation (9) is as follows,

n [ 2 e 3 a ( − 1 ) − 2 e a ( 1 ) / 2 a e a ( e 2 a + ( 1 − e − ƛ x i sinh a ( 1 − e − ƛ x i ) / cosh a ( 1 − e − ƛ x i ) = √ / Π η e ^ − ( y − δ ) 2 / 2 η 2 d y (10)

Solving for the final price, a, in Equation (10) yields the maximum gain from short selling. Target stock is then, purchased, with exponential increase in satisfaction as prices rise, and gains increase. An exponential utility function is assumed, which intersects with the Poisson jump process to provide first the threshold price to qualify for incentive compensation, and then, the optimal price to

earn profits. The omega ratio (risk measure) is elevated to the fourth power to account for kurtosis.

For a pair of securities, that are independent and identically distributed, the joint moment of expected values is,

E [ X i X j ] = 1 / ( n − j + 1 ) ƛ E [ X i ] + E [ X j 2 ] (11)

where, ƛ = rate parameter, i < j , [

The right side of Equation (11) is equated to the Ito solution to the Poisson jump process.

1 / ( n − j + 1 ) ƛ E [ X i ] + E [ X j 2 ] = ∂ g / ∂ t + μ ∂ g / ∂ S + σ 2 / 2 ∂ 2 g / ∂ S 2 + h ( t ) ∫ _ Δ g ( g η g ( . ) d Δ g ) d t + ∂ g / ∂ S σ d W ( t ) + d J _ g ( t ) (12)

S includes the drift, and jump, g ( S ( t ) , t ) = function of the jump process, d S ( t ) .

h ( t ) = stochastic process

Taking the first derivative of both sides,

1 / ( n − j + 1 ) ƛ E [ X i ] + E [ X j 2 ] = ∂ 2 g / ∂ t 2 + μ ∂ 2 g / ∂ S 2 + σ 2 / 2 ∂ 3 g / ∂ S 3 + h ( t ) ( g η g ( . ) Δ g ) + ∂ 2 g / ∂ S 2 σ d 2 W ( t ) + d 2 J _ g ( t ) (13)

Taking the second derivative of Equation (12),

1 / ( n − j + 1 ) ƛ E [ X i ] + E [ X j 2 ] = ∂ 3 g / ∂ t 3 + μ ∂ 3 g / ∂ S 3 + σ 2 / 2 ∂ 4 g / ∂ S 4 + h ( t ) ( g 2 η g ( . ) Δ g ) + ∂ 3 g / ∂ S 3 σ d 3 W 3 ( t ) + d 3 J _ g (t)

Omitting higher moments, and adding the omega ratio, Ώ , and H, the threshold for minimum risk, the solution for E(X_{i}) and E(X_{j}), yields the expected optimal price in Equation (14).

x + H + ( 1 − Ώ ( H ) ) 4 ( H − x ) + 1 / ( n − j + 1 ) ƛ E [ X i ] + E [ X j 2 ] = ∂ 3 g / ∂ t 3 + μ ∂ 3 g / ∂ S 3 + h ( t ) ( g 2 η g ( . ) Δ g ) + ∂ 3 g / ∂ S 3 σ d 3 W 3 ( t ) + d 3 J _ g ( t ) (14)

After the threshold has been reached, traders engage in herding. Traders from multiple locations purchase target stock concurrently. As trade prices > the bid/ask midpoint, market makers sense a surge in demand. They increase ask prices, continuing to increase them, until the ask price = bid/ask midpoint, and trading ceases, The surge in prices of target stock, from collective demand by a large number of traders may be modeled by an Esscher transformation, which is frequently used to measure collective risk [

The Esscher-transformed martingale measure, multiplies the right side of Equation (14), to magnify the impact of collective risk. Omitting the omega ratio function, which applies to the reaching of the threshold, Equation (14) transforms to

1 / ( n − j + 1 ) ƛ E [ X i ] + E [ X j 2 ] = [ ∂ 3 g / ∂ t 3 + μ ∂ 3 g / ∂ S 3 + h ( t ) ( g 2 η g ( . ) Δ g ) + ∂ 3 g / ∂ S 3 σ d 3 W 3 ( t ) + d 3 J _ g ( t ) ] [ exp { t ( H B + 0.5 h 2 σ 2 ) + ∫ ( e h ( e x − 1 ) 1 e x − 1 ≤ 1 } − x 1 { x ≤ 1 } ( x ) ) v d x ) } (15)

h 2 = constant, σ 2 = variance of returns, v d x = martingale process

OP represents the exponential utility function, which intersects with the Poisson Jump Process at Point A and Point B, to yield optimal target prices. Another hallmark of hedge fund trading is timing ability. When hedge fund traders complete short selling, they observe that the minimum price of the stock has not been achieved, since the uptick rule restricts short selling to two rounds. Therefore, on the day after announcement, traders drive prices to a minimum, with successive rounds of put buying. With put purchases, traders sell at high prices, market makers reduce bid prices, traders purchase at lower prices, taking gains, then sell again, taking gains at each round, until the minimum stock price is reached. Empirically. [

d / d x = [ ( ∂ 4 g / ∂ t 4 + μ ∂ 4 g / ∂ S 4 + h ( t ) ( g 3 η g ( . ) Δ g ) + ∂ 4 g / ∂ S 4 σ d 4 W 4 ( t ) + d 4 J _ g ( t ) ] [ { t ( H B + 0.5 h 2 σ 2 ) − 1 + ( e h ( e x − 1 ) 1 e x − 1 ≤ 1 } − 1 ) v ) } → 0 (16)

As exponential gains are expected, put buyers’ utility functions follow an exponential distribution. However, there is uncertainty of earning put gains, as we do not know when the lower bound will be reached. The strengthening desire for put gain, coupled with uncertainty suggests the need for a Poisson jump process with stochastic integrals. Stochastic integrals account for collective uncertainty from all hedge fund traders, engaged in trading options on stock with underlying values. To further accommodate uncertainty, we include measures of kurtosis, for 2 random variables, X, and Y, of put option prices, to account for fat-tailed distributions. The pricing distribution for a Poisson jump process with stochastic integrals, together with kurtosis is shown in Equation (17). The cokurtosis between X and Y is an order 4 tensor [

exp ( ƛ ∫ ∫ ( e y f t − 1 ) v ( d y ) d t ) + 1 / σ X + Y 4 ( σ X 4 K u r t [ X ] + 4 σ X 3 σ Y C o k u r t [ X , X , X , Y ] + 6 σ X 2 σ Y 2 C o k u r t [ X , X , Y , Y ] + 4 σ X σ Y 3 C o k u r t [ X , Y , Y , Y ] + σ Y 4 K u r t [ Y ] ) (17)

Taking first derivatives of Expression (17),

( ƛ ∫ ∫ ( e y f t − 1 ) v ( d y ) d t ) − 1 + 4 / σ X + Y 3 ( 4 σ X 3 4 S k [ X ] + 12 σ X 2 σ Y C o s k [ X , X , X , Y ] + 12 σ Y C o s k [ X , X , Y , Y ] + 12 σ X σ Y 2 C o s k [ X , Y , Y , Y ] + 16 σ Y 3 S k [ Y ] ) (18)

Taking second derivatives of Expression (17),

( ƛ ∫ ( e y f t − 1 ) v ( d y ) d t ) − 1 + 12 / σ X + Y 2 ( 12 σ X 2 12 V a r [ X ] + 24 σ X σ Y C o v a r [ X , X , X , Y ] + 12 σ Y C o v a r [ X , X , Y , Y ] + 24 σ X σ Y C o v a r [ X , Y , Y , Y ] + 48 σ Y 2 3 V a r [ Y ] ) (19)

Equating the left side of Equation (15) with Equation (19) yields Equation (20), whose solution at X is the lower bound put option price.

1 / ( n − j + 1 ) ƛ E [ X i ] + E [ X j 2 ] = ( ƛ ∫ ( e y f t − 1 ) v ( d y ) d t ) − 1 + 12 / σ X + Y 2 ( 12 σ X 2 12 V a r [ X ] + 24 σ X σ Y C o v a r [ X , X , X , Y ] + 12 σ Y C o v a r [ X , X , Y , Y ] + 24 σ X σ Y C o v a r [ X , Y , Y , Y ] + 48 σ Y 2 3 V a r [ Y ] ) (20)

Global Macro Strategies. [

0.5 + 0.5 ( s g n ) ( x − μ ) ( 1 − exp ) ( − | x − μ | / b ) = ∂ g / ∂ t + μ ∂ g / ∂ S + σ 2 / 2 ∂ 2 g / ∂ S 2 + h ( t ) ∫ _ Δ g ( g η g ( . ) d Δ g ) d t + ∂ g / ∂ S σ d W ( t ) + d J _ g ( t ) (21)

where μ = scale parameter, and b = location parameter of the Laplace distribution Taking derivatives of both sides of Equation (21),

0.5 ( s g n ) ( 1 − μ ) + ( | x − μ | / b ) = ∂ 2 g / ∂ t 2 + μ ∂ 2 g / ∂ S 2 + σ 2 / 2 ∂ 3 g / ∂ S 3 + h ( t ) ( g 2 η g ( . ) Δ g ) + ∂ 2 g / ∂ S 2 σ d 2 W ( t ) + d 2 J _ g ( t ) (22)

Taking second derivatives of both sides of Equation (21),

0.5 ( s g n ) ( 1 − μ ) + ( | 1 − μ | / b ) = ∂ 3 g / ∂ t 3 + μ ∂ 3 g / ∂ S 3 + σ 2 / 2 ∂ 4 g / ∂ S 4 + h ( t ) ( g η g ( . ) Δ g ) + 3 g / ∂ S 3 σ d 3 W / d W 3 + d 3 J + d 3 g ( t ) (23)

As more traders purchased technology stocks, market makers increased ask prices, so that trade prices continually occurred above the bid-ask midpoint. Herding ensued, with synchronization of trades. A trader at one location, would place a large purchase order, with two other traders at two other locations, placing similar orders of the same size, concurrently. After six years of meteoric rise in technology prices, [_{1} > the first derivative of the pricing distribution at time, t_{2} > the first derivative of the pricing distribution at time, t_{3}.

∂ 2 g / ∂ t 1 2 + μ ∂ 2 g / ∂ S 2 + σ 2 / 2 ∂ 3 g / ∂ S 3 + h ( t 1 ) ( g 2 η g ( . ) Δ g ) + ∂ 2 g / ∂ S 2 σ d 2 W ( t 1 ) + d 2 J _ g ( t 1 ) > ∂ 2 g / ∂ t 2 2 + μ ∂ 2 g / ∂ S 2 + σ 2 / 2 ∂ 3 g / ∂ S 3 + h ( t 2 ) ( g 2 η g ( . ) Δ g ) + ∂ 2 g / ∂ S 2 σ d 2 W ( t 2 ) + d 2 J _ g ( t 2 ) > ∂ 2 g / ∂ t 3 2 + μ ∂ 2 g / ∂ S 2 + σ 2 / 2 ∂ 3 g / ∂ S 3 + h ( t 3 ) ( g 2 η g ( . ) Δ g ) + ∂ 2 g / ∂ S 2 σ d 2 W ( t 3 ) + d 2 J _ g ( t 3 ) (24)

Other traders, who may not have invested in the same technology stocks, relied on sell signals for their stocks, which were not forthcoming. Short-sale constraints prevented short selling for gain, with falling prices. Such investors continued to hold their technology portfolios, losing heavily upon market correction. Therefore, short sale constraints impose an upper bound to gaining from falling prices, as shown in Equation (25). The expression for short sale constraints, i.e. ( S e − P ) ∗ V = 0 , where S_{e} = short sale price, P = purchase price of shorted stock, V = volume. intersects with Equation (23).

0.5 ( s g n ) ( 1 − μ ) + ( | 1 − μ | / b ) = ∂ 3 g / ∂ t 3 + μ ∂ 3 g / ∂ S 3 + σ 2 / 2 ∂ 4 g / ∂ S 4 + h ( t ) ( g η g ( . ) Δ g ) + 3 g / ∂ S 3 σ d 3 W / d W 3 + d 3 J + d 3 g ( t ) = ( S e − P ) ∗ V = 0 (25)

This section has described directional hedge fund strategies, whereby traders herd to derive gains from corporate events through corporate event-driven strategies, and macroeconomic phenomena, termed global/macroeconomic events. It involves direct, timely action to influence the direction of events.

Commodity Trading on Yen Futures and New Zealand Dollar Futures. A currency futures contract is valued as the sum of its spot price (current price), and term premium (with price fluctuations during the lengthy delivery period). A non-directional hedge fund trader may borrow in yen by shorting yen futures with about 2% interest rates, and then, invest in New Zealand dollar futures with high 8.25% rates of return, gaining on the interest rate differential. This strategy is non-directional hedge fund investing, in that it employs an investment opportunity in commodities in two currencies, using timing to know when to exit

B is the price at which directional hedge fund traders sold to take gains. Some short selling occurred at point C. Those traders who did not predict the end of the bubble, retained stock, valued at point D. Source: This Paper.

short selling, and commence investing in NZ dollar futures. It does not use herding, to influence the direction of price movements. The utility to the hedge fund manager arises from satisfaction from a shorting-cum-investing strategy, with returns based on yield differentials between the two currencies, rather than active trading. It follows that there is gradual increase in satisfaction, which may be modeled by a lognormal distribution. The value of yen futures is based on its current spot price, which in turn, depends upon the central bank’s policy of price stability, near-zero interest rates, liquidity as the third largest reserve currency in the world, and a strong trade balance with large and increasing trade inflows into Japan. [

y 1 ⋅ ( σ 1 2 Π ) − 1 exp ( − ( ln y 1 − μ 1 ) 2 2 σ 1 2 ) = x 1 − 1 ⋅ ( σ 2 2 Π ) − 1 exp ( − ( ln x 1 − μ 2 ) 2 / 2 σ 1 2 ) (26)

where y , σ 1 , μ 1 pertain to the utility function, and x 1 , σ 2 , μ 2 pertain to the yen.

y 1 ⋅ ( σ 1 2 Π ) − 1 ( ( ln y 1 − μ 1 ) / σ 1 2 ) 2 = x 1 − 1 ⋅ ( σ 2 2 Π ) − 1 ( ( ln x 1 − μ 2 ) / σ 2 2 ) 2 (27)

[

y 1 ⋅ ( σ 1 2 Π ) − 1 exp ( − ( ln y 1 − μ 1 ) 2 / 2 σ 1 2 ) = x 1 − 1 ⋅ ( σ 2 2 Π ) − 1 exp ( − ( ln x 1 − μ 2 ) 2 / 2 σ 1 2 ) + x 2 − 1 ( 1 + r ) ⋅ ( σ 3 2 Π ) − 1 exp ( − ( ln x 2 − μ 3 ) 2 / 2 σ 3 2 ) + x 3 − 1 ( 1 + r ) ⋅ ( σ 4 2 Π ) − 1 exp ( − ( ln x 3 − μ 4 ) 2 / 2 σ 4 2 ) (28)

Funds obtained from shorting yen futures may be invested in high-yield New Zealand dollar futures. [

y 1 ⋅ ( σ 1 2 Π ) − 1 exp ( − ( ln y 1 − μ 1 ) 2 / 2 σ 1 2 ) = Y 2 − a 2 − b 2 d 2 X 2 / d X 2 2 (29)

where y , σ 1 , μ 1 pertain to the utility function, and Y 2 is the price, a 2 = constant, b 2 = change in NZ dollar futures for a unit change in NZ dollar risk, delivery period, only.

[

y 1 ⋅ ( σ 1 2 Π ) − 1 exp ( − ( ln y 1 − μ 1 ) 2 / 2 σ 1 2 ) = ( Y 3 − a 3 − b 3 d 2 X 3 / d X 3 2 − Y 2 − a 2 − b 2 d 2 X 2 / d X 2 2 ) + ( Y 4 − a 4 − b 4 d 2 X 4 / d X 4 2 − Y 3 − a 3 − b 3 d 2 X 3 / d X 3 2 ) + ⋯ (30)

where Y 3 , a 3 , b 3 , and X 3 are the term premium variables for the third period, where Y 4 , a 4 , b 4 , and X 4 are the term premium variables for the fourth period,

Adding the second derivative of the spot price from the Short Roll strategy in Equation (28) to the right side of Equation (30), yields the total optimal NZ dollar futures price, considering both current and future prices.

y 1 ⋅ ( σ 1 2 Π ) − 1 exp ( − ( ln y 1 − μ 1 ) 2 / 2 σ 1 2 ) = x 1 − 1 ⋅ ( σ 2 2 Π ) − 1 exp ( − ( ln x 1 − μ 2 ) 2 / 2 σ 1 2 ) + x 2 − 1 ( 1 + r ) ⋅ ( σ 3 2 Π ) − 1 exp ( − ( ln x 2 − μ 3 ) 2 / 2 σ 3 2 ) + x 3 − 1 ( 1 + r ) ⋅ ( σ 4 2 Π ) − 1 exp ( − ( ln x 3 − μ 4 ) 2 / 2 σ 4 2 ) + ( Y 3 − a 3 − b 3 d 2 X 3 / d X 3 2 − Y 2 − a 2 − b 2 d 2 X 2 / d X 2 2 ) + ( Y 4 − a 4 − b 4 d 2 X 4 / d X 4 2 − Y 3 − a 3 − b 3 d 2 X 3 / d X 3 2 ) + ⋯ (31)

Convertible Arbitrage. [

2 η / C 0 2 ( η ) F L ( η , ρ ) [ ln 2 ρ + q L ( η ) / ρ L ( η ) ] + 2 / ( 2 L + 1 ) − L ∑ α L L ( η ) ρ k + L = x 1 − 1 ⋅ ( σ 2 2 Π ) − 1 exp ( − ( ln x 1 − μ 2 ) 2 / 2 σ 1 2 ) (32)

Taking first derivatives of both sides,

2 η / C 0 2 ′ ( η ) F ′ L ( η , ρ ) [ 1 / 2 ρ + q ′ L ( η ) / ρ ′ L ( η ) ] + 2 / ( 2 L + 1 ) − L α L L ( η ) ρ k + L = x 1 − 2 ⋅ ( σ 2 2 Π ) − 2 exp ( − ( ln X 1 − μ 2 ) 2 / 2 σ 1 2 )

Taking second derivatives of Equation (32), yields the optimal short sale price,

2 η / C 0 2 ′ ′ ( η ) F ″ L ( η , ρ ) [ − 1 / 2 ρ 2 + q ″ L ( η ) / ρ ″ L ( η ) ] + 2 / ( 2 L + 1 ) − L α L L ′ ( η ) ρ k + L = 2 x 1 − 3 ⋅ ( σ 2 2 Π ) − 3 ( − ( 1 / X 1 − μ 2 ) 2 / 2 σ 1 2 ) (33)

The convertible bond is converted to growth stocks. The investment in growth stocks requires an expectation of rising utility (satisfaction) with rising prices. Yet, the expectations of growth will not be as rapid as before a merger, suggesting that a controlled increase in utility, with rising prices, such as a Bessel function, be employed to describe the utility function of the non-directional trader (see Equation (34)).

= ( 0.5 z v ) ∑ k = 0 ∞ [ ( − 0.25 z 2 ) k / k ! Γ ( v + k + 1 ) ] (34)

where z = a plane cut along the negative real axis, v = the first derivative of the Bessel function, k = constant

Taking first derivatives of Equation (34).

= ( 0.5 z v ) [ ( − 0.25 z 2 ) k / k ! Γ ( v + k + 1 ) ]

Taking second derivatives of Equation (34),

= ( 0.5 v z v − 1 ) [ ( − 0.25 z ) k / k ! Γ ′ ( 1 + k ) ] (35)

Equating Equation (35)’s Bessel utility function with the second derivative of the Poisson jump process,

( 0.5 v z v − 1 ) [ ( − 0.25 z ) k / k ! Γ ′ ( 1 + k ) ] = ∂ 3 g / ∂ t 3 + μ ∂ 3 g / ∂ S 3 + σ 2 / 2 ∂ 4 g / ∂ S 4 + h ( t ) ( g η g ( . ) Δ g ) + 3 g / ∂ S 3 σ d 3 W / d W 3 + d 3 J + d 3 g ( t ) (36)

Solving for z in Equation (36), yields the optimal stock price, for maximum gain.

Non-directional hedge fund strategies seek entrepreneurial opportunities for profit-making, using timing to purchase at low prices and sell at high prices. These strategies demonstrated a sophisticated understanding of the direction of market movements, like those of directional hedge fund traders. They are distinguished from directional hedge fund strategies, in that they do not employ herding to influence the direction of price movements.

Sector Funds Theory. Mutual fund traders are liquidity traders. Liquidity traders

emphasize the minimization of risk, as specified by the [

R j = R f + σ j / σ M ⋅ ( R M − R f ) (37)

where R j = return on the stock, σ j = risk of biotechnology stock, R M − R f

σ M = market risk, R M − R f = market risk premium,

The risk of the biotechnology stock increases portfolio risk substantially, so that maintaining a diversified portfolio is used to reduce this risk to just above the market risk, shown in

The utility function for the mutual fund manager may be expressed as the following Legendre integral, ST, which intersects with the Capital Market Line at point F, in

( 1 − z 2 ) d 2 / d z 2 2 z ⋅ d w / d z + [ v ( v + 1 ) μ 2 / ( 1 − z 2 ) ] w = 0 (38)

With degree v, and order μ, with singularities at ±1, α, as ordinary branch points―μ, v, arbitrary complex constants. Z = x + 1, y, z are real numbers in the interval −1 ≤ +1. The Arrow-Pratt risk aversion is multiplied by the left side of Equation (38), and the right side is equated to the Capital Market Line, which acts as the pricing distribution,

( 1 − z 2 ) d 2 / d z 2 2 z ⋅ d w / d z + [ v ( v + 1 ) μ 2 / ( 1 − z 2 ) ]

Taking second derivatives of Equation (38), and multiplying by the Arrow-Pratt Risk aversion, and equating to the second derivative of the Capital Market Line, to yield the optimal price,

{ ( 1 − z 2 ) d 4 / d z 4 2 z ⋅ d 3 w / d z 3 + [ v ( v + 1 ) μ 2 / ( 1 − z 2 ) ] d 2 / d x 2 w } / { ( 1 − z 2 ) d 3 / d z 3 2 z d 2 w / d z 2 + [ v ( v + 1 ) μ 2 / ( 1 − z 2 ) ] d w / d x } = R j − R f − d / d x ( σ j / σ M ) ⋅ ( R M − R f ) (39)

Large Cap US Growth Fund Theory. A mutual fund may identify itself as a large capitalization, US Growth Fund, which invests in large market capitalization, US stocks with growth rates among the leading 20% of growth rates of all domestic stocks traded on the S & P 500, or large market capitalization index. Style investing is followed, with a Markowitz portfolio created, in accordance with Equation (39). Stocks are added to the portfolio if their standard deviation (risk) remains within 1 standard deviation of that of the market portfolio, or the S & P 500. This portfolio will definitely locate on the Capital Market Line, as it does not have undue risk to lift its risk values above the Capital Market Line. An additional measure of risk may be applied from the Capital Asset Pricing Model, an extension of the Markowitz conceptualization. This is the stock’s beta coefficient, the variation of stock returns with market portfolio returns. Given that the market portfolio’s beta coefficient has a value of 1, the average stock in this mutual fund should have a beta from 0.8 - 1.2, or within a narrow band at or about 1. Systematic risk levels may not rise beyond 1.2. Non-systematic risk, measured by firm-specific risk, is likely to be minimal, given the minimization of covariance risk by holding highly diversified portfolios of hundreds of stocks. Given the nature of mutual funds, fund traders adhere to the philosophy of earning returns, just above the market return, while maintaining risk with the aforementioned 0.8 - 1.2 beta band.

There is no attempt to seek out investment opportunities with potential for future growth. There is no timing of purchases or sales to take advantages of preferable pricing. There is also no herding, as traders act individually, purchasing for their own inventories, rather than through collective action. They employ a buy-and-hold strategy, purchasing at any time. Since there are a number of large capitalization stocks, mutual fund traders may be purchasing continuously. They hold the portfolio for a long period of time, selling only if the risk of any stock rises beyond 1.2. After a certain period, specified by the investing philosophy of the employer, the securities are sold, and gains or losses assessed. [

An inverse sine function [

1 / ( m + n ) ⋅ sinh m + 1 z cosh n − 1 z + ( n − 1 ) / ( m + n ) ∫ sinh m z cosh n − z d z = R j − R f − σ j / σ M ⋅ ( R M − R f ) (40)

where, m, n = constants, z = random variable, specifying varying levels of satisfaction with risky investments, Taking the first derivative of Equation (40),

1 / ( m + n ) ⋅ cosh m + 1 − z sinh n − 1 z + ( n − 1 ) / ( m + n ) sinh m z cosh n − z d z = d R j / d x − R f − σ ′ j / σ ′ M ⋅ ( d R M / d x − R f ) (41)

where R f is assumed to be a constant Taking the second derivative of Equation (40),

( m + 1 ) / ( m + n ) ⋅ sinh m z ( n − 1 ) cosh n z + m ( n − 1 ) / ( m + n ) cosh m − 1 − z sinh n − 1 d z = d 2 R j / d x 2 − R f − σ ″ j / σ ″ M ⋅ ( d 2 R M / d x 2 − R f ) (42)

Emerging Markets Mutual Fund. The final mutual fund portfolio that will be considered is a portfolio of emerging markets securities. [

( 1 − z 2 ) d 2 / d z 2 2 z ⋅ d w / d z + [ v ( v + 1 ) μ 2 / ( 1 − z 2 ) ] w = 0 (43)

With degree v, and order μ, with singularities at ±1, α, as ordinary branch points―μ, v, arbitrary complex constants. Z = x + 1, y, z real number is the interval −1 ≤ +1.

The most appropriate pricing distribution is described by the International Capital Asset Pricing Model. The Capital Asset Pricing Model, which describes returns on domestic securities is an extension of the Capital Market Line, which separates total security risk into market risk, or the stock’s sensitivity to market conditions, such as unemployment reports, and firm-specific risk found in the residuals, which is diversified. The International Capital Asset Pricing model, adds an additional item to assist in the valuation of global securities, β_{1} (R_{fo} − R_{d}) where β_{1} = sensitivity of domestic returns to changes in foreign currencies, and (R_{fo} − R_{d}) = the additional return demanded by investors for investing in a foreign portfolio.

The Arrow-Pratt risk aversion is multiplied by the left side of Equation (43), and the right side is equated to the International Capital Asset Pricing Model (ICAPM) which acts as the pricing distribution,

( 1 − z 2 ) d 2 / d z 2 2 z ⋅ d w / d z + [ v ( v + 1 ) μ 2 / ( 1 − z 2 ) ]

Taking second derivatives of Equation (43), and multiplying by the Arrow-Pratt Risk aversion, and equating to the second derivative of the International Capital Asset Pricing Model to yield the optimal price,

{ ( 1 − z 2 ) d 4 / d z 4 2 z ⋅ d 3 w / d z 3 + [ v ( v + 1 ) μ 2 / ( 1 − z 2 ) ] d 2 / d x 2 w } / { ( 1 − z 2 ) d 3 / d z 3 2 z d 2 w / d z 2 + [ v ( v + 1 ) μ 2 / ( 1 − z 2 ) ] d w / d x } = R j − R f − d / d x ( σ j / σ M ) ⋅ ( R M − R f ) + d / d x β 1 ( R f o − R d ) (44)

representation of its pricing distribution.

In summary, mutual fund strategies are dictated by corporate policy. Mutual fund traders trade for liquidity, i.e. to purchase and retain securities for long-term gains. They do not pursue entrepreneurial opportunities, or time the market or herd, to drive security price movements. They are usually forbidden from short selling, or undertaking high-risk investing.

Both the [

Future research could develop equations to predict price movements in the event of more stringent regulatory restrictions on herding. Herding frequently occurs during price run-ups on the day before dividend announcements, positive earnings announcements, and on target stock in mergers. At this time, regulatory restrictions on herding activity are minimal, with any civil penalties being well within the financial capabilities of cash-rich hedge funds. Regulators may also monitor multimarket trading. Upon being subjected to short-sale constraints in the stock market, traders shift trading to the relatively less regulated options market, in a form of multimarket trading. Trading could be halted if regulators feel that hedge funds are driving prices excessively downwards, thereby destabilizing the market. Regulators may also be concerned that mutual fund traders, with the narrowest profit margins, are paying high transaction fees. Market makers typically increase transaction fees to uninformed mutual fund traders, to neutralize their losses to informed hedge fund traders.

How long can mutual fund managers continue to follow corporate practice and engage in low risk, market return-based trading ? The advent of the Internet has ushered in the instantaneous transmission of news throughout the world. Millenial investors, exposed to successful high-risk investing may abhor long-term, stable returns in favor of riskier equity investments. They may even consider very high risk-investments, such as foreign currency derivatives, or commodity options. This shift in risk preferences may lead mutual fund traders to engage in the arbitrage strategies of nondirectional hedge funds. It is unlikely that they would have the social connections to engage in herding. Future research should investigate shifts in risk tolerance of mutual fund traders.

Additional investigations could also trace directional hedge fund utility movements and price changes for events that generate negative signals, such as bankruptcy, reorganization, or merger deal failure, using the expressions developed in this paper. Research should also develop theoretical formulations for directional hedge fund activity during global macroeconomic events, such as Brexit, currency crises when a currency, or group of currencies lose up to 20% of their value within a few days, tariffs, or the global sell-off of stocks in December 2018. Non-directional hedge fund strategies may need to display superior timing abilities. For example, the paper has developed the formulation for the commodity trading strategy of short selling yen futures, using the proceeds to invest in New Zealand dollar futures. Future research should create formulations for a higher-risk commodity trading strategy of short selling yen futures, using the proceeds to invest in volatile Mexican pesos. As the Mexican peso is subject to sudden, unexpected devaluations, the trading formulation should first short yen, then invest in Mexican pesos, followed by timing to convert the Mexican pesos to yen prior to Mexican peso devaluation.

As volatility-based trading increases, traders may become less risk-averse, so that the risk-averse trader may become an anachronism. Research should ascertain if there is a mainstream reduction in risk aversion in society. Perhaps, we are seeing a fundamental shift in attitudes to risk, from risk aversion to risk-taking, given shifts in the global environment, including disruptive technology, tariffs, and slow growth in the industrialized west. As the shapes of utility functions will change, this study must be repeated for utility functions based on new statistical distributions.

This paper has created a novel framework for hedge fund investing, and mutual fund investing. The three components of directional hedge fund investing include seeking out entrepreneurial opportunities, timing, and herding. This framework has not been addressed in prior literature. Non-directional hedge funds engage in the seeking of entrepreneurial opportunities, and timing of purchases and sales. However, they do not engage in herding. For directional hedge funds, this paper has set forth the equations showing the rapid increase in prices and abnormal returns earned at optimal prices, from herding.

Both of the hedge fund strategies require that traders demonstrate analytical abilities in comprehending market and security price movements. Mutual fund traders do not require superior analytical skills with buy-and-hold strategies, based on benchmarks created by their employers. The philosophy of following corporate practice prevails, even if the benchmark changes, such as different market benchmarks for sector funds, US growth funds, and international funds. Future research should assess whether the growing uncertainties of future economic environments may be met with established corporate practices in mutual fund investing. Possibly, riskier, and more innovative investing strategies will be more profitable than simple buy-and-hold.

This paper has fulfilled its objective of creating complex utility functions. Hitherto, utility functions have consisted of power utility functions, lognormal distribution, and hypergeometric distributions. We have presented a richer array of functions, including Bessel functions, hyperbolic cosine distributions, and Legendre integrals, Laplace distributions, quadratic utility functions, inverse sine distributions, and exponential distributions. Our utility functions distinguish between hedge funds and mutual funds, on the basis of risk aversion. Directional hedge fund traders are risk-takers. They believe in herding to influence prices, earning abnormal returns. Non-directional hedge fund traders are more risk-averse than hedge fund traders, eschewing market destabilization through herding. Mutual fund traders are the most risk-averse, making small volumes of trades, regardless of market conditions. Directional hedge fund traders make the highest gains with market opportunities, with mutual fund traders making the lowest gains. During the technology bubble, directional hedge fund traders sold early, achieving price gains, but no losses. Non-directional hedge fund traders made price gains, though some gains were lost, by not selling early. Mutual fund traders experienced the most losses, as they ignored market signals, owning securities that were rapidly losing value.

The Arrow-Pratt measure of risk aversion was created over fifty years ago, for developed countries. Perhaps, there are new measures of risk aversion, particularly for emerging markets, which are distinct from the markets in developed countries Emerging markets have early stage infrastructure development, and 7% - 9% growth rates, as opposed to established infrastructure, and 2% - 3% growth rates in developed countries. Wealth expectations that determine risk aversion in emerging markets may, thus, depend on a different set of macroeconomic variables. Another adjustment to risk aversion could be a measure for risk-taking that is closer to gambling for directional hedge fund traders, as initial success from risky investment stimulates the desire for further risk-taking [

The author declares no conflicts of interest regarding the publication of this paper.

Abraham, R. (2019) Hedge Fund Investing or Mutual Fund Investing: An Application of Multi-Attribute Utility Theory. Theoretical Economics Letters, 9, 605-632. https://doi.org/10.4236/tel.2019.94042